Abstract
A Lawson-type exponential integrator combined with the Fourier pseudo-spectral method is provided for the nonlinear Double Sine-Gordon equation (DSGE), while the nonlinearity is characterized by \(\beta /\epsilon \) with small parameter \(\epsilon \in (0,1]\) and interaction parameter \(\beta \in (0,+\infty )\). In comparison to the Sine-Gordon equation, DSGE has many properties of solitons as well as its own unique new features. This is the first work to numerically simulate the physical phenomena arising from DSG kinks collisions. The improved uniform error bounds are proved by using the regularity compensation oscillatory (RCO) technique, which are \(O(\alpha ^2\tau +h^m)\) up to the long time at \(T_{\epsilon }=T/\alpha ^2\), where \(\alpha =\epsilon \) for \(\beta \ge 1\) and \(\alpha ={\epsilon }/{\beta }\) for \(0<\epsilon<\beta <1\). Based on the uniform bounds, the error estimation for the discrete energy is derived. Furthermore, the improved error bounds are extended to two oscillatory DSGEs with \(O(\epsilon ^2)\) and \(O(\epsilon ^2/\beta ^2)\) wavelength in time. Numerical examples are provided to illustrate the accuracy and discrete energy property of the proposed method.
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References
Aktosun, T., Demontis, F., Cornelis, V.D.M.: Exact solutions to the Sine-Gordon equation. J. Math. Phys. 51(12), 1262 (2010)
Atanasova, P.K., Boyadjiev, T., Shukrinov, Y.M., Zemlyanaya, E.: Numerical modeling of long josephson junctions in the frame of double Sine-Gordon equation. Math. Models Comput. Simul. 3(3), 389–398 (2011)
Bao, W., Feng, Y., Yi, W.: Long time error analysis of finite difference time domain methods for the nonlinear Klein-Gordon equation with weak nonlinearity. Commun. Comput. Phys. 26(5), 1307–1334 (2019)
Bao, W., Cai, Y., Feng, Y.: Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity. SIAM J. Numer. Anal. 60(4), 1962–1984 (2022)
Bao, W., Cai, Y., Feng, Y.: Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation. Math. Comput. 92(341), 1109–1139 (2023)
Bin, H., Qing, M., Yao, L., Weiguo, R.: New exact solutions of the double Sine-Gordon equation using symbolic computations. Appl. Math. Comput. 186(2), 1334–1346 (2007)
Bullough, R., Caudrey, P., Gibbs, H.: The double Sine-Gordon equations: A physically applicable system of equations. In: Solitons, vol 17, Springer, pp 107–141 (1980)
Dikandé, A., Kofané, T.: Double Sine-Gordon solitons in two-dimensional crystals. J. Phys. Condens Mat. 7(10), L141 (1995)
Domairry, G., Davodi, A.G., Davodi, A.G.: Solutions for the double Sine-Gordon equation by exp-function, tanh, and extended tanh methods. Numer Methods Partial Differential Equations 26(2), 384–398 (2010)
Duckworth, S.: Radiative Processes in Multi-Level and Degenerate Atomic Systems. The University of Manchester (United Kingdom) (1976)
Duckworth, S., Bullough, R.K., Caudrey, P.J., Gibbon, J.D.: Unusual soliton behaviour in the self-induced transparency of q(2) vibration-rotation transitions. Phys. Lett. A 57(1), 19–22 (1976)
Fang, D., Zhang, Q.: Long-time existence for semi-linear Klein-Gordon equations on tori. J. Differential Equations 249(1), 151–179 (2010)
Faou, E., Schratz, K.: Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime. Numer. Math. 126(3), 441–469 (2014)
Feng, Y.: Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity. Numer Methods Partial Differential Equations 37(1), 897–914 (2021)
Feng, Y., Schratz, K.: Improved uniform error bounds on a lawson-type exponential integrator for the long-time dynamics of Sine-Gordon equation. (2022) DOI: https://doi.org/1048550/arXiv:221109402
Feng, Y., Yi, W.: Uniform error bounds of an exponential wave integrator for the long-time dynamics of the nonlinear Klein-Gordon equation. Multiscale Model. Simul. 19(3), 1212–1235 (2021)
Feng, Y., Maierhofer, G., Schratz, K.: Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations. Math. Comput. 93(348), 1569–1598 (2024)
Gani, V.A., Marjaneh, A.M., Askari, A., Belendryasova, E., Saadatmand, D.: Scattering of the double Sine-Gordon kinks. Eur. Phys. J. C 78, 345 (2018)
Hong, B.: New jacobi elliptic function solutions for the double Sine-Gordon equation. World J. Model 7(2), 133–138 (2009)
Zagrodziński, J.: Particular solutions of the Sine-Gordon equation in 2 + 1 dimensions. Phys. Lett. A 72(4–5), 284–286 (1979)
Szeftel, J.: Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres. Int. Math. Res. Not IMRN 2004(37), 1897–1966 (2004)
Kasai, Y., Tanda, S., Hatakenaka, N., Takayanagi, H.: Fluxon dynamics in isolated long josephson junctions. Physica C 352(1–4), 211–214 (2001)
Kitchenside, P.W.: The double Sine-Gordon equation: Numerical and perturbative studies. PhD thesis, University of Manchester Institute of Science and Technology (UMIST) (1979)
Leibbrandt, G.: New exact solutions of the classical Sine-Gordon equation in 2 + 1 and 3 + 1 dimensions. Phys. Rev. Lett. 41(7), 435–438 (1978)
Li, J., Zhu, L.: A uniformly accurate exponential wave integrator fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials. Numer Algorithms 92(2), 1367–1401 (2023)
Mussardo, G., Riva, V., Sotkov, G.: Semiclassical particle spectrum of double Sine-Gordon model. Nuclear Phys. B 687(3), 189–219 (2004)
Nikolay, K.: Breather and soliton wave families for the Sine-Gordon equation. Proc. Rco. Lond A. 454(1977), 2409–2423 (1998)
Peng, Y.Z.: Exact solutions for some nonlinear partial differential equations. Phys. Lett. A 314(5–6), 401–408 (2003)
Peyravi, M., Montakhab, A., Riazi, N., Gharaati, A.: Interaction properties of the periodic and step-like solutions of the double Sine-Gordon equation. Eur. Phys. J. B 72, 269–277 (2009)
Popov, S.P.: Influence of dislocations on kink solutions of the double Sine-Gordon equation. Comput. Math. Math. Phys. 53(12), 1891–1899 (2013)
Takács, G., Wágner, F.: Double Sine-Gordon model revisited. Nuclear Phys. B 741(3), 353–367 (2006)
Wang, H., Fu, Y.: Exact traveling wave solutions for the (2+1)-dimensional double Sine-Gordon equation using direct integral method. Appl. Math. Lett. 146(108), 798 (2023)
Wang, M., Li, X.: Exact solutions to the double Sine-Gordon equation. Chaos, Solitons Fractals 27(2), 477–486 (2006)
Wang, Y., Zhao, X.: A symmetric low-regularity integrator for nonlinear Klein-Gordon equation. Math. Comput. 91(337), 2215–2245 (2022)
Wazwaz, A.M.: The tanh method: exact solutions of the Sine-Gordon and the sinh-gordon equations. Appl. Math. Comput. 167(2), 1196–1210 (2005)
Wazwaz, A.M.: The tanh method and a variable separated ode method for solving double Sine-Gordon equation. Phys. Lett. A 350(5–6), 367–370 (2006)
Wu, Y., Yao, F.: A first-order fourier integrator for the nonlinear Schrödinger equation on \(\mathbb{T} \) without loss of regularity. Math. Comput. 91(335), 1213–1235 (2022)
Acknowledgements
The authors would like to express their appreciation to the anonymous reviewer for the invaluable comments that have greatly improved the quality of the manuscript. This work was supported by Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2024A1515012548) and the National Natural Science Foundation of China (Nos. 12471374, 12171148).
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Ling Zhang: Formal analysis, Data Curation, Software, Validation, Writing-Original Draft. Huailing Song: Conceptualization, Methodology, Writing Reviewing & Editing, Project Administration, Supervision. Wenfan Yi: Formal analysis, Validation, Writing Reviewing & Editing
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Appendices
The proof of Lemma 1
By the definition of the function f and g, we have
According to (9)and (10), we can write the local truncation error as
where \(\mathscr {Q}\) is defined in (22), \(\mathscr {R}^n=K_1+K_2+K_3\), and the corresponding representations are as follows
We can reformulate \(K_1\) as
then we obtain that under the assumptions \(\mathrm (A)\) with \(m\ge 0\),
where \(\alpha =\epsilon \) for \(\beta \ge 1\) and \(\alpha ={\epsilon }/{\beta }\) for \(\epsilon<\beta <1\). Similarly, by the definitions of the functions \(\mathscr {H}(\psi )\), \(r_1(\psi )\) and \(r_2(\psi )\), we can obtain estimates for \(K_2\) and \(K_3\) as \(\Vert K_2 \Vert _{1}\lesssim \tau ^2\alpha ^4\) and \(\Vert K_3 \Vert _{1}\lesssim \tau ^6\alpha ^4\), respectively. Combining the above results, we have
Obviously, (22) can be reformulated as
By the properties of the operator \(e^{i\sigma \langle \partial _x \rangle }\), we have
From the above results, under the assumption \(\mathrm (A)\) we have that
Then, we complete the proof of the local error bounds (23).
The proof of Theorem 3
Firstly we give the following lemma [25].
Lemma B.2
Given \(u,v \in X_M\) and \(\tilde{u}_l,\tilde{v}_l\) defined in (13), we have
From Lemma B.2 and the energy expression (31), we get
then the discrete energy deviation from the initial energy is
where
where \(\zeta ^n\) and \(U^n\) are the values of the exact solution \(\partial _t u(x,t)\) and u(x, t) of DSGE(4) at \(t_n\), respectively, \(\Vert {z^0}-\zeta ^0\Vert _{0}^2=0\) and \(\Vert {u^0}-U^0\Vert _{1}^2=0\).
Applying inequality \(1-\cos (u)\le u^2/2\), we can get
Under the assumption (A), combining the estimates of \(E_1\) and \(E_2\), we can obtain
since the exact solution of the DSGE (4) satisfies conservation of energy, therefore
according to the error estimation results (1), we have the following estimates for the discrete energy:
In addition, if the exact solution is sufficiently smooth, e.g., \(u,\partial _t u \in H^{\infty }\), the estimate for the discrete energy deviation for sufficiently small \(\tau \) is
The proof of (32) is completed.
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Zhang, L., Song, H. & Yi, W. Improved Uniform Error Bounds on a Lawson-type Exponential Integrator Method for Long-Time Dynamics of the Nonlinear Double Sine-Gordon Equation. J Sci Comput 102, 25 (2025). https://doi.org/10.1007/s10915-024-02752-6
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DOI: https://doi.org/10.1007/s10915-024-02752-6
Keywords
- Double Sine-Gordon equation
- Long-time dynamics
- Lawson-type exponential integrator
- Regularity compensation oscillatory
- Improved uniform error bounds